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Distributions to Estimate Population Parameter

Distributions to Estimate Population Parameter

The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...

Prediction Intervals

Prediction Intervals

The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y.

Interpretation of Confidence Intervals

Interpretation of Confidence Intervals

A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
Confidence intervals have confidence coefficients that are crucial for their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99, which can be written as percentages–90%, 95%, and 99%, respectively.
Suppose a person calculates a confidence interval with a confidence coefficient of 0.95. In that case, they can...

Statistical Inference Techniques in Hypothesis Testing: Parametric Versus Nonparametric Data

Statistical Inference Techniques in Hypothesis Testing: Parametric Versus Nonparametric Data

Statistical inference techniques, paramount in hypothesis testing, differentiate into two broad categories: parametric and nonparametric statistics.
Parametric statistics, as the name suggests, assumes that data follow a specific distribution, often a normal distribution. This assumption enables robust hypothesis testing and estimation. Parametric methods, like the Student's t-test or Goodness-of-fit test, are frequently employed in biostatistics due to their robustness. For instance,...

Estimating Population Mean with Unknown Standard Deviation

Estimating Population Mean with Unknown Standard Deviation

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the...

Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...

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Approximate Bayesian Inference.

Pierre Alquier¹

¹Center for Advanced Intelligence Project (AIP), RIKEN, Tokyo 103-0027, Japan.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

This summary is machine-generated.

This editorial introduces a special issue on Approximate Bayesian Inference. It covers methods and applications for complex statistical modeling and data analysis.

Keywords:

Bayesian statistics Gibbs posterior Langevin Monte Carlo Laplace approximations Markov chain Monte Carlo PAC-Bayes approximate Bayesian computation expectation-propagation machine learning sequential Monte Carlo variational approximations

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Area of Science:

Statistics
Machine Learning
Computational Science

Background:

Approximate Bayesian Inference (ABI) methods are crucial for intractable posterior distributions.
The special issue addresses the growing need for efficient and scalable Bayesian computation.

Discussion:

Explores various ABI techniques, including variational inference, Markov Chain Monte Carlo (MCMC) methods, and particle filters.
Discusses the theoretical underpinnings and practical implementation challenges of these methods.

Key Insights:

Highlights advancements in approximate Bayesian inference for large-scale and high-dimensional problems.
Emphasizes the importance of rigorous evaluation and comparison of different ABI approaches.

Outlook:

Predicts future research directions in approximate Bayesian inference, focusing on automation and deep learning integration.
Suggests potential impacts on fields like artificial intelligence, bioinformatics, and econometrics.